Combinatorial Optimization

Körte, Bernhard; Vygen, Jens

doi:10.1007/978-3-662-21711-5

Cited by 250 publications

(199 citation statements)

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“…All the above problems are polynomial-time solvable (see, e.g., [23]) in their unbudgeted version (k = 0), but become NP-hard [1,6] even for a single budget constraint (k = 1). For the case of one budget (k = 1), PTASs are known for spanning tree [35] (see also [21]), 1 The assumption that k is a constant is crucial in this paper, since many of the presented algorithms will have a running time that is exponential in k, but polynomial for constant k. 2 We recall that E is a finite ground set and I ⊆ 2 E is a nonempty family of subsets of E (independent sets) which have to satisfy the following two conditions: (i) I ∈ I, J ⊆ I ⇒ J ∈ I and (ii) I, J ∈ I, |I| > |J| ⇒ ∃z ∈ I \ J : J ∪ {z} ∈ I.…”

Section: Introductionmentioning

confidence: 99%

New approaches to multi-objective optimization

et al. 2013

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A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Many classical optimization problems, such as maximum spanning tree and forest, shortest path, maximum weight (perfect) matching, maximum weight independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems efficient deterministic and randomized approximation schemes are known. Not much is known however about the case of two or more budgets: filling this gap, at least partially, is the main goal of this paper. In more detail, we obtain the following main results:• Using iterative rounding for the first time in multi-objective optimization, we obtain multi-criteria PTASs (which slightly violate the budget constraints) for spanning tree, matroid basis, and bipartite matching with k = O(1) budget constraints.• We present a simple mechanism to transform multi-criteria approximation schemes into pure approximation schemes for problems whose feasible solutions define an independence system. This gives improved algorithms for several problems. In particular, this mechanism can be applied to the above bipartite matching algorithm, hence obtaining a pure PTAS.• We show that points in low-dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for k-budgeted matroid independent set. • We present a deterministic approximation scheme for k-budgeted matching (in general graphs), where k = O(1). Interestingly, to show that our procedure works, we rely on a non-constructive result by Stromquist and Woodall, which is based on the Ham Sandwich Theorem.

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Section: Introductionmentioning

confidence: 99%

New approaches to multi-objective optimization

et al. 2013

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“…To transfer money between nodes, we have edges from sources to sinks. This naturally results in a bipartite flow network, typical for the so called Hitchcock Problem [9].…”

Section: The Monetary Flow Networkmentioning

confidence: 99%

“…Secondly, one can use specialized algorithms for the Minimum Cost Flow Problem. For an overview see, e.g., the corresponding chapter in [9].…”

Section: Theorem 2 Let L ⊆ B Be a Solution If P Is A Profit Sharingmentioning

confidence: 99%

Modeling profit sharing in combinatorial exchanges by network flows

et al. 2013

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In this paper we study the possibilities of sharing profit in combinatorial procurement auctions and exchanges. Bundles of heterogeneous items are offered by the sellers, and the buyers can then place bundle bids on sets of these items. That way, both sellers and buyers can express synergies between items and avoid the well-known risk of exposure (see, e.g., Cramton, Shoham, & Steinberg, Combinatorial Auctions, 2006). The reassignment of items to participants is known as the Winner Determination Problem (WDP). We propose solving the WDP by using a Set Covering formulation, because profits are potentially higher than with the usual Set Partitioning formulation, and subsidies are unnecessary. The achieved benefit is then to be distributed amongst the participants of the auction, a process which is known as profit sharing. The literature on profit sharing provides various desirable criteria. We focus on three main properties we would like to guarantee: Budget balance, mean ing that no more money is distributed than profit was generated, no negative transfers, which guarantees to each player that participation does not lead to a loss, and the core property, which provides every subcoalition with enough money to keep them from separating. We characterize all profit distributions that satisfy these three conditions by a monetary flow network and establish a connection to the famous VCG payment scheme (Clarke in Public Choice 18:19ff, 1971; Groves in Econometrica 41:617-631, 1973; Vickrey in J. Finance 16:8-37, 1961) and the Shapley Value (Shapley in Kuhn & Tucker (Eds.), Contributions to the Theory of Games, vol. II, pp. 307-317, 1953). Finally, we introduce a novel profit sharing scheme based on scaling down VCG discounts using max-min fairness principles to achieve budget balance. It approximates the Shapley Value linearly in the number of bundles and can be computed efficiently

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“…Among others we mention linear production games (Owen, 1975), minimum cost spanning tree games (Granot and Huberman, 1981) Matching in graphs are combinatorial optimization problems. Because of its importance they have been studied in depth (Lovász and Plummer, 1986;Korte and Vygen, 2000). In a pioneering paper, Shapley and Shubik (1972) analyze the bipartite graph case as a cooperative problem.…”

Section: Introductionmentioning

confidence: 99%

Insights into the Nucleolus of the Assignment Game

2015

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Abstract:We show that the family of assignment matrices which give rise to the same nucleolus form a compact join-semilattice with one maximal element, which is always a valuation (see p.43, Topkis (1998)). We give an explicit form of this valuation matrix. The above family is in general not a convex set, but path-connected, and we construct minimal elements of this family. We also analyze the conditions to ensure that a given vector is the nucleolus of some assignment game.JEL Codes: C71.

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Combinatorial Optimization

Cited by 250 publications

References 0 publications

New approaches to multi-objective optimization

New approaches to multi-objective optimization

Modeling profit sharing in combinatorial exchanges by network flows

Insights into the Nucleolus of the Assignment Game

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